Math CentralQuandaries & Queries


Question from Neda, a parent:

when dividing a number which is greater than 1 by a number that is less than 1, will the quotient be greater than or less than the dividend? Explain.

We have two responses for you.


Did you try it? (I assume these are all positive numbers.) Take some positive number greater than 1 (the dividend) and divide it by a positive number less than 1 (the divisor). What do you get for an answer (the quotient)? For example take a dividend of 10 and divided by 1/2. What do you get as the quotient? Write this as an equation. Multiplication is the inverse of division so rewrite this equation as a multiplication equation. Does this help answer your question?



Greater, provided the divisor is positive.

It would be more accurate and clearer to say that when dividing a number which is greater than zero by a number between 0 and 1, then the quotient is greater than the dividend.

If the divisor is negative and the dividend is positive, then the quotient is negative (and hence less than the dividend).

Ways to see this:

"Extending the trend": For positive x, x/3 < x/2 < x/1, so we would expect x/(0.5) to be still bigger. This is not rigorous but is at least suggestive.

Inverse to multiplication: Let 0<a<1, x>0. Then x/a * a = x. Multiplying by a makes any positive number smaller, so x/a is bigger than x.

Good Hunting!



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